Journal of Combinatorial Theory, Series B  TB1695 journal of combinatorial theory, Series B 67, 325343 (1996) article no. 0049

On Inequivalent Representations of over Finite Fields* James Oxley and Dirk Vertigan

Department of Mathematics, Louisiana State University, Baton Rouge, Louisiana 70803-4918

and

Geoff Whittle

Department of Mathematics, Victoria University, P.O. Box 600, Wellington, New Zealand Received May 30, 1995

Kahn conjectured in 1988 that, for each prime power q, there is an integer n(q) such that no 3-connected GF(q)-representable has more than n(q) inequivalent GF(q)-representations. At the time, this conjecture was known to be true for q=2 and q=3, and Kahn had just proved it for q=4. In this paper, we prove the conjecture for q=5, showing that 6 is a sharp value for n(5). Moreover, we also show that the conjecture is false for all larger values of q.  1996 Academic Press, Inc.

1. INTRODUCTION

In the study of representations of matroids over finite fields, the problem of inequivalent representations arises almost immediately. For example, consider the 9-point -3 matroid M whose only non-trivial lines are three disjoint 3-point lines. For a large enough F, the matroid M can be represented by a set of points in which the non-trivial lines are copunctual, and can also be represented by a set of points in which they are not; see Fig. 1. Now, automorphisms of projective planes preserve copunctuality, so it is clear that the two representations of M are inequivalent for any natural notion of equivalence of representations. For small enough fields, this problem does not arise. It is easily seen that GF(2)-representations of a matroid are equivalent, and Brylawski and Lucas [4] have shown that GF(3)-representations are unique. Kahn [9] proved that GF(4)-representations are unique for 3-connected matroids.

* This paper is dedicated to Don Row who introduced all three authors to matroids. 325 0095-8956Â96 18.00 Copyright  1996 by Academic Press, Inc. All rights of reproduction in any form reserved.

File: 582B 169501 . By:CV . Date:31:07:96 . Time:15:09 LOP8M. V8.0. Page 01:01 Codes: 3973 Signs: 2041 . Length: 50 pic 3 pts, 212 mm 326 OXLEY, VERTIGAN, AND WHITTLE

Figure 1

However, for q>4, one cannot guarantee uniqueness of representations, even for 3-connected matroids. Given this, one could at least hope that, for such fields, the number of inequivalent representations was limited in some way. Indeed, in [9], Kahn conjectured that, for each prime power q, there is an integer n(q) such that no 3-connected GF(q)-representable matroid has more than n(q) inequivalent representations. In this paper, we prove Kahn's conjecture for q=5 showing that there are at most six inequivalent GF(5)-representations of a 3-connected matroid. We also provide counter- examples to show that Kahn's conjecture is false for all larger values of q. Kahn's proof that GF(4)-representations of 3-connected matroids are unique uses an elegant geometric argument. A crucial result needed in this argument is the following theorem of Seymour [19]. A matroid N uses a set X if X E(N).

Theorem. (1.1) If x1 and x2 are elements of a 3-connected nonbinary matroid M, then M has a U2, 4 -minor using [x1 , x2]. Our proof of Kahn's conjecture for GF(5) is structured similarly to Kahn's proof for GF(4). Section 3 is devoted to proving a result analogous to Theorem 1.1. The crucial fact needed is that if M is a 3-connected GF(5)- representable matroid with a U2, 4-restriction and a U2, 5-minor, then M has a U2, 5-minor using the elements of the U2, 4-restriction. This fact, stated as Corollary 3.10, follows immediately from more general theorems proved in Section 3. Both Theorem 1.1 and the results of Section 3 are examples of ``roundedness'' results. A brief discussion of such results and their role in matroid theory is included in Section 2. The proof of the conjecture for GF(5), given in Section 4, then applies Corollary 3.10, using

File: 582B 169502 . By:SD . Date:28:06:96 . Time:07:57 LOP8M. V8.0. Page 01:01 Codes: 2290 Signs: 1775 . Length: 45 pic 0 pts, 190 mm ON INEQUIVALENT REPRESENTATIONS 327 a similar geometric argument to Kahn's proof for GF(4). Not surprisingly, some complications arise which are peculiar to the GF(5) case. The counterexamples to Kahn's conjecture for q7 are given in Section 5. We present two classes of counterexamples. The first class deals with the case where the multiplicative group of GF(5) has a proper subgroup of order at least three. The only cases not covered by the first class of counter- examples occur when q=2k for some k>2 such that 2k&1 is prime. But, in these cases, the additive group of the field has a proper subgroup of order at least three. The second class of counterexamples includes this case. It is of interest to ask if the results in this paper have any implications for a longstanding conjecture of Rota [14] that the set of forbidden minors for representations over GF(q) is finite. This conjecture has only been proved when q # [2, 3] and, recently [7], when q=4, and all known proofs in these cases make essential use of unique representability [1, 8, 10, 17, 21, 22]. We do not know if there is any connection between the results of this paper and the validity of Rota's conjecture in general. However, it is clear that any proof of Rota's conjecture for a prime power q>5 would require significantly different techniques from those that have been used to prove the conjecture for q4.

2. PRELIMINARIES

Familiarity is assumed with the elements of matroid theory. Throughout, the terminology follows [12]. We include here a brief discussion of those aspects of matroid theory that are of particular importance for this paper. Representations A matroid M is representable over a field F or F-representable if there is a function . from the ground set of M into a V over F which preserves rank. In other words, rM(X )=rV[.(x): x # X] for all X E(M). Equivalently, M is F-representable if and only if there is a A over F whose columns are labelled by the elements of M such that, for all X E(M), the matrix consisting of the columns of A that are labelled by elements of X has rank equal to rM(X ). Such a matrix is called a represen- tation of M. It is clear that a matroid is F-representable if and only if the associated simple matroid is F-representable. Thus, when considering representability questions, one frequently restricts attention to simple matroids, and we shall do this from now on. In this case, the function . above may be viewed as a one-to-one function into a projective space over F. Indeed, it is commonplace to think of a rank-r simple F-representable matroid M as being embedded in the projective space PG(r&1, F), that is, as being a

File: 582B 169503 . By:CV . Date:31:07:96 . Time:15:09 LOP8M. V8.0. Page 01:01 Codes: 3116 Signs: 2655 . Length: 45 pic 0 pts, 190 mm 328 OXLEY, VERTIGAN, AND WHITTLE

restriction of PG(r&1, F). Two such embeddings .1 and .2 are equivalent if there is an automorphism % of PG(r&1, F) such that %(.1(e))=.2(e) for all e in E(M). Let |E(M)|=n. Since there is a natural way to associate such an embedding with each r_n matrix representation of M over F, this defines equivalence of two such matrix representations. This definition can be translated into purely matrix terms as follows. First, for r # [1, 2], the automorphism group of PG(r&1, F) is the symmetric group. Thus if r(M)=r2, then all r_n matrix representations of M over F are equiv- alent. If r(M)>2, it follows from a theorem, sometimes known as the Fundamental Theorem of Projective Geometry, that two r_n matrix representations are equivalent if one can be obtained from the other by a sequence of the following operations. (For details, see [12, Section 6.3].) (i) Interchange two rows. (ii) Multiply a row by a non-zero member of F. (iii) Replace a row by the sum of that row and another. (iv) Interchange two columns (moving their labels with the columns). (v) Multiply a column by a non-zero member of F. (vi) Replace each entry of the matrix by its image under some automorphism of F. We say that M is uniquely representable over F if all r_n representations of M over F are equivalent. The fact that, by the above definition, all representations of a rank-2 matroid are equivalent has the disconcerting consequence that a matroid M and its dual may have different numbers of inequivalent representations, although this can only occur if r(M)orr(M*) is 2. One could remedy this by modifying the definition so that, for representations of rank-2 matroids to be equivalent, they must be obtainable from each other via a sequence of operations (i)(vi). Such a change does not alter the results of this paper, and we prefer to follow Kahn [9] and maintain the link between equivalence and automorphisms of the underlying projective geometry. Roundedness For a positive integer t, a class N of matroids is t-rounded if every member of N is (t+1)-connected and the following condition holds: If M is a (t+1)-connected matroid having an N-minor and X is a subset of E(M) with at most t elements, then M has an N-minor using X. In this terminology, Seymour's theorem (1.1) amounts to saying that [U2, 4] is 2-rounded. The task of determining whether a given class of matroids is t-rounded is potentially infinite. However, Seymour [16, 20] has shown that, when t is 1 or 2, this task is finite (see also [12, Theorem 11.3.9]).

File: 582B 169504 . By:CV . Date:31:07:96 . Time:15:09 LOP8M. V8.0. Page 01:01 Codes: 3123 Signs: 2529 . Length: 45 pic 0 pts, 190 mm ON INEQUIVALENT REPRESENTATIONS 329

(2.1) Theorem. Lettbe1or 2 and N be a collection of (t+1)-connected matroids. Then N is t-rounded if and only if the following condition holds: If M is a (t+1)-connected matroid having an N-minor N such that |E(M)&E(N)|=1, and X is a subset of E(M) with at most t elements, then M has an N-minor using X. It is natural to extend the notion of roundedness and look, not just at the size of subsets, but at their matroid structure as well. Indeed, Reid [13] has already introduced the notion of ``triangle roundedness'' and his idea can be further generalized. The results in Section 3 can be interpreted in terms of such a generalization. One could say that they are results on

``U2, n-roundedness''. In this light, Theorem 3.1 can be interpreted as an analogue of Theorem 2.1, and Theorem 3.2 amounts to the assertion that + + [U2, 5 , F7 ] is ``U2, 4 -rounded'' where F7 is obtained from the Fano matroid by freely adding an element on one of the lines.

3. SOME ROUNDEDNESS RESULTS

In this section, we prove some roundedness results, one of which, Corollary 3.10, plays a crucial role in the proof of Kahn's conjecture for q=5.

Theorem. (3.1) Let M be a 3-connected matroid having a U2, n+1-minor and a subset X such that M | X$U2, n. Then one of the following holds.

(a) M has a U2, n+1-minor using X. (b) For some r in [3, 4], M has a 3-connected rank-r minor N using

X such that N has a U2, n+1-minor and |E(N)|=2n+r&3. When n4, this theorem can be strengthened. In particular, when n is 2 or 3, alternative (b) can be eliminated. This is elementary when n=2; for n=3, it follows without difficulty from Theorem 1.1. For the remainder of this section, we assume that n4. Our strengthening of Theorem 3.1 when + n=4 involves the matroid F7 that is obtained from the Fano matroid + F7 by freely adding an element on one of the lines. Clearly F 7 has a U2, 5 -minor.

Theorem (3.2) . Let M be a 3-connected matroid having a U2, 5-minor and a subset X such that M | X$U2, 4. Then M has a minor N using X such + that N is isomorphic to U2, 5 or F7 . Since Theorem 3.2 is a strengthening of a case of Theorem 3.1, much of the proofs of the two theorems will be common. The next four lemmas

File: 582B 169505 . By:CV . Date:31:07:96 . Time:15:09 LOP8M. V8.0. Page 01:01 Codes: 2887 Signs: 2179 . Length: 45 pic 0 pts, 190 mm 330 OXLEY, VERTIGAN, AND WHITTLE will be used in both proofs. In each of these, M will satisfy the following condition:

(3.3) M is a 3-connected matroid having a U2, n+1-minor and a subset

X such that M | X$U2, n. Moreover, M is a minor-minimal matroid that has these properties but has no U2, n+1-minor using X. Certainly if M satisfies (3.3), then r(M)3. Let the elements of X be labelled by x1 , x2 , ..., xn .

(3.4) Lemma. Suppose that M satisfies (3.3). Then X is a modular line of M.

Proof. The matroid M is 3-connected but has no U2, n+1-minor using X,soXis certainly a flat of M. Hence X is a line of M. Assume that X is not modular. Then, by a well-known characterization of modular flats (see, for example, [3, Theorem 3.3] or [12, Proposition 6.9.2]), M has a hyper- plane Y that avoids X. Clearly r(X _ Y )=r(M). Thus a 2-element subset

[xi, xj] of X can be extended to a basis for M by adding some set Z consisting of r(M)&2 elements of Y. Consider MÂZ. This has rank 2 and contains X.Ifxand x$ are distinct elements of X, then rMÂZ([x, x$])=2, for otherwise Z _ [x, x$] has rank at most r(M)&1 and yet spans the basis Z _ [xi, xj] of M. Since rMÂZ(Y&Z)=1, we may choose an element y of Y&Z that is not a loop of MÂZ. Then y is not parallel to an element of X in MÂZ, for otherwise Y & X contains this element. We conclude that the restriction of MÂZ to X _ y is an (n+1)-point line; a contradiction. K

(3.5) Lemma. Suppose that M satisfies (3.3). If H is a hyperplane of M that does not contain X, then |E(M)&(H_X)|2. Proof. By Lemma 3.4, X is a modular line, so X must meet H. Since H does not contain X, it follows that |X & H|=1, so |X&H|=n&13. If E(M)&(H_X) is empty, then [H, X&H] is a 2-separation of M;a contradiction. Thus we may assume that E(M)&(H_X) contains some element e. We may also assume that e is the only such element, for otherwise the lemma holds. Therefore r(H)+r(X&H)&r(M"e)=1, so [H, X&H] is a 2-separation of M"e. Suppose this 2-separation is minimal. Then |H|=2. But |H & X |=1, so |E(M)&X|=2. Thus M has a 2-element cocircuit; a contradiction. Hence the 2-separation [H, X&H] is non- minimal. Therefore, by Bixby [2] (see also [12, Proposition 8.4.6]), MÂe t has no non-minimal 2-separations, and its simplification, MÂe, is 3-connected. Now |X & H |=1. Without loss of generality, we may assume that t X & H=[x ]. Since e  X, we may also assume that X E(MÂe). Certainly t n t (MÂe)|X$U . Since MÂe is also 3-connected, the choice of M implies t 2, n that MÂe has no U2, n+1-minor.

File: 582B 169506 . By:CV . Date:31:07:96 . Time:15:09 LOP8M. V8.0. Page 01:01 Codes: 3280 Signs: 2442 . Length: 45 pic 0 pts, 190 mm ON INEQUIVALENT REPRESENTATIONS 331

Let X$=[x1, x2, ..., xn&1]. Then M has X$ _ e as a cocircuit. Thus M"(X$&x ) has [x , e] as a cocircuit. Hence M"(X$&x )Âx $ 1 t1 1 1 M"(X$&x1)Âe. Since MÂe has no U2, n+1-minor, M"(X$&x1)Âx1 has no

U2, n+1-minor. But, in MÂx1 , the elements x2 , x3 , ..., xn are in parallel. Thus MÂx1 has no U2, n+1-minor. Similarly, none of MÂx2 ,

MÂx3 , ..., MÂxn&1 has a U2, n+1-minor. It follows that, for every U2, n+1- minor M1 of M, there is a subset T of X$ such that M1 uses X$&T, and

M1 is a minor of M"T. Evidently, if M has a U2, n+1-minor using at least two elements of X, then M has a U2, n+1-minor using X. Thus U2, n+1 is a minor of M"T for some (n&2)-element subset T of X$. But M"T has (X$&T)_e as a 2-element cocircuit. Hence M"TÂe has a U -minor. t 2, n+1 This contradicts the fact that MÂe has no U2, n+1-minor. K

(3.6) Lemma. If M satisfies (3.3), then |E(M)|r(M)+2n&3.

Proof. Since M has a U2, n+1-minor, we have U2, n+1 $M"YÂZ for some independent set Z and some coindependent set Y.If|Y|n&2, then

r*(M)n&2+r*(U2, n+1)=n&2+n&1, and so |E(M)|r(M)+2n&3, and the lemma holds. Thus we may assume that |Y |n&3. But M has no U2, n+1-minor using two or more elements of X,so|X&(Y_Z)|1. Hence |X & Y |+|X&Z|n&1. Therefore, as |X & Y |n&3, we deduce that |X & Z|2. It follows, since r(X )=2 and Z is independent, that |X & Z|=2. Thus |X&Z|=n&2. Since every element of X&Z is a loop of MÂ(X & Z), we deduce that U2, n+1 must occur as a minor of MÂ(X & Z)"(X&Z). Hence

r*(M)|X&Z|+r*(U2, n+1)=n&2+n&1, and, as before, we conclude that |E(M)|r(M)+2n&3. K

(3.7) Lemma. If M satisfies (3.3), then |E(M)|2n+1. Proof. Lemma 2.8 of Coullard and Reid [5] describes the structure of a minor-minimal 3-connected matroid M1 that is a minor of M using some 3-element subset X$ofE(M) and has a U2, n+1-minor. If X$ is a circuit

[x1, x2, x3] of M, then, as in Reid [13], it is straightforward to deduce that, up to a permutation of [x1, x2, x3], one of the following occurs:

(i) |E(M1)|(n+1)+3;

(ii) there is an element f of E(M1)&[x1, x2, x3] such that either [x1, x2, f] or [x1, x2 , x3 , f] is a cocircuit of M1 , and |E(M)|= (n+1)+4;

File: 582B 169507 . By:CV . Date:31:07:96 . Time:15:09 LOP8M. V8.0. Page 01:01 Codes: 3033 Signs: 2065 . Length: 45 pic 0 pts, 190 mm 332 OXLEY, VERTIGAN, AND WHITTLE

(iii) there is an element f of E(M1)&[x1, x2, x3] such that

[x1, x2, f]is a circuit of M1 , and |E(M1)|=(n+1)+4.

As M1 is a minor of M, we can write M1=M"UÂV for some subsets U and V of E(M) where V is independent and U is coindependent. Clearly t V & X is empty. Let M2=M"(U&X )ÂV. Certainly M2 is 3-connected and this matroid may be labelled so that its ground set contains X. It follows t from the fact that M satisfies (3.3) that M2 =M2=M. Thus (U&X ) _ V is empty, and so M1=M"(U & X ) and r(M1)=r(M).

If (i) occurs, then the lemma certainly holds. If (ii) occurs, then M1" f, and hence M" f, is the union of X and a hyperplane; a contradiction to Lemma 3.5. Thus we may assume that (iii) occurs. Then f # X, for otherwise

M has an (n+1)-point line using x1 , x2 , ..., xn , and f. Therefore |E(M)|=

|E(M2)||E(M1)|+n&4=2n+1. K

Proof of Theorem 3.1. Let M be a minor-minimal matroid that satisfies the hypotheses of the theorem but has no U2, n+1-minor using X. Then M satisfies (3.3) and so, on combining Lemmas 3.6 and 3.7, we deduce that

r(M)+2n&3|E(M)|2n+1.

Therefore either

(i) r(M)=4 and |E(M)|=2n+1=2n+r&3; or (ii) r(M)=3 and |E(M)|=2n=2n+r&3; or (iii) r(M)=3 and |E(M)|=2n+1.

To complete the proof of Theorem 3.1, we shall show that (iii) does not occur. Assume the contrary. Since M has a U2, n+1-minor but has no such minor using X, there is an element x of X for which MÂx has a U2, n+1- minor. In MÂx, the n&1 elements of X&x are in parallel. Thus

MÂx"(X&[x, x$]) has a U2, n+1-minor where x$#X&x. The matroid MÂx"(X&[x, x$]) has exactly n+2 elements and rank 2. Since it has a

U2, n+1-minor, it must be isomorphic to either U2, n+2 or the matroid that is obtained from U2, n+1 by adding an element in parallel to one of the elements. In the latter case, let [y,y$] be the unique 2-circuit of the matroid and, in the former case, let y be any element of the U2, n+2-minor other than x$. In each case, M" y has a U2, n+1-minor and has a U2, n - restriction using X. Thus the choice of M implies that M" y is not 3-con- nected. But r(M" y)=3. Therefore every element of M" y lies on one of X and another line, L say. Hence M has a hyperplane, clM(L), such that

|E(M)&(X_clM(L))|1. This is a contradiction to Lemma 3.5. K

File: 582B 169508 . By:CV . Date:31:07:96 . Time:15:09 LOP8M. V8.0. Page 01:01 Codes: 2992 Signs: 2200 . Length: 45 pic 0 pts, 190 mm ON INEQUIVALENT REPRESENTATIONS 333

We now know that Theorem 3.1 holds. To complete the proof of Theorem 3.2, we need an additional lemma on matroids satisfying (3.3) in the special case that n=4.

(3.8) Lemma. Suppose that n=4 and that M satisfies (3.3). Then r(M){4. Proof. Assume that r(M)=4. Then, by Lemmas 3.6 and 3.7,

|E(M)|=9. Let E(M)&X=Y. Now M"UÂV$U2, 5 for some 2-element independent set V and some 2-element coindependent set U. Consider M | Y.As[Y, X] is not a 2-separation of M, we deduce that r(M | Y )=4. Moreover, if M | Y has a coloop y, then clM(Y&y)isa hyperplane of M and |E(M)&(X_clM(Y&y))|=1; a contradiction to

Lemma 3.5. Hence M | Y$U4, 5. Clearly M"UÂV contains at most one element of X.So |(V_U)&X|3. But |U|=2. Therefore |V & X |1. Now suppose that |V & X |=1. Then V & Y contains a unique element, say y. The rank-3 matroid MÂy has a U -minor. Since X is a flat of M t 2, 5 avoiding y, the matroid MÂy may be chosen so that its ground set contains t X. Therefore the choice of M implies that MÂy is not 3-connected. But

(MÂy)|(Y&y)$U3, 4 so, in MÂy, two of the elements of Y&y must be parallel to elements of X. Hence |clM(X _ y)|=7, so M has a 2-cocircuit; a contradiction. We may now suppose that |V & X |>1. Since |V & X ||V|=2, we conclude that |V & X |=2. Therefore the two elements of X&V are loops of MÂV and so U=X&V. Moreover, no two elements of Y are parallel in MÂV. Now each 3-element subset of Y spans a different plane of M. Moreover, by Lemma 3.4, each of these planes meets X. Since |Y|=5, there are ten such planes. But |X |=4, so some point p of X lies on at least three of these planes. Choose three such planes. It is routine to show that two of these planes share two common elements of Y. The intersection of these two planes has rank at most two. Thus there is a line of M that contains p and two elements of Y. But these two elements are parallel in MÂV; a contra- diction. K Proof of Theorem 3.2. Let M be a minor-minimal matroid satisfying the hypotheses of the theorem but having no U2, 5 -minor using X. Then M satisfies (3.3) so, by Theorem 3.1 and Lemma 3.8, r(M)=3 and |E(M)|=8. Now X is a modular line of M and, for some element x of X, the matroid

MÂx has a U2, 5 -minor. But MÂx has seven elements and rank 2 and has X&x as a parallel class. Therefore each U2, 5 -minor of MÂx is obtained from it by deleting any two of the three elements of X&x. Thus X is the

File: 582B 169509 . By:CV . Date:31:07:96 . Time:15:09 LOP8M. V8.0. Page 01:01 Codes: 2927 Signs: 2369 . Length: 45 pic 0 pts, 190 mm 334 OXLEY, VERTIGAN, AND WHITTLE only dependent line of M containing x. Since M is 3-connected, M"X is isomorphic to U3, 4 or U2, 3 ÄU1, 1. In the latter case, the four distinct lines of M that contain at least two elements of E(M)&X must meet the line X in different points. Hence x is on two dependent lines of M; a contra- diction. We conclude that M"X$U3, 4 . Moreover, every line of M that is spanned by two elements of E(M)&X must meet X in some element of X&x. The next lemma, the straightforward proof of which is omitted, immediately implies that M"x$F7 .

(3.9) Lemma. The only 7-element rank-3 simple matroid whose ground set can be partitioned into a 4-circuit and a modular line is F7 .

+ Since F7 is the unique simple matroid that is obtained by adding an + element to one of the lines of F7 , we conclude that M$F7 . Hence Theorem 3.2 is proved. K

Since F7 is representable only over fields of two, it follows + that F7 is not representable over GF(5). The following corollary, which now follows immediately from Theorem 3.2, is crucial for the proof of Kahn's conjecture for GF(5).

(3.10) Corollary. Let M be a 3-connected GF(5)-representable matroid having a U2, 5 -minor and a subset X such that M | X$U2, 4 . Then M has a U2, 5 -minor using X. At this stage, it is natural to ask for which fields the analogue of Corollary 3.10 holds. While this question is perhaps an aside, it has an interesting answer, for such an analogue holds only when q # [2, 3, 4, 5]. These are exactly the values of q for which Kahn's conjecture holds. In other words, we have the following:

(3.11) Proposition. Let q be a prime power exceeding 5. Then there is aGF(q)-representable matroid M having a U2, q-minor and a subset X such that M | X$U2, q&1 but such that M has no U2, q -minor using X. This proposition follows from examples presented at the end of Section 5. These examples are deferred to that section since they are related to the counterexamples to Kahn's conjecture for q>5.

4. PROOF OF KAHN'S CONJECTURE FOR GF(5)

(4.1) Theorem. A 3-connected matroid has at most six inequivalent representations over GF(5).

File: 582B 169510 . By:CV . Date:31:07:96 . Time:15:10 LOP8M. V8.0. Page 01:01 Codes: 2799 Signs: 2095 . Length: 45 pic 0 pts, 190 mm ON INEQUIVALENT REPRESENTATIONS 335

Proof. Assume that the theorem fails and let M be a minor-minimal counterexample. Certainly M must be GF(5)-representable. Suppose that M is ternary. Whittle [23, Corollary 2.8] showed that a 3-connected ternary matroid has at most q&2 inequivalent representations over GF(q) for any prime power q>2. It follows that M has at most three inequivalent representations over GF(5). Hence we may suppose that M is non-ternary. Thus, by the excluded-minor characterization of ternary matroids [1, 17],

M has U2, 5 , U3, 5 , F7 ,orF*7 as a minor. The last two matroids are representable only over fields of characteristic two. Thus M has U2, 5 or

U3, 5 as a minor. By Oxley [11, Theorem 1.6] (see also [12, Proposi- tion 11.2.16]), a 3-connected matroid with rank and corank at least three has a U2, 5 -minor if and only if it has a U3, 5 -minor. Thus either M has a U2, 5 -minor, or M has a U3, 5-minor and r*(M)=2. In the latter case, since

M is 3-connected, M$Un, n+2 for some n3. In that case, since M is GF(5)-representable, n is 3 or 4. We conclude that either M has a U2, 5 - minor, or M is isomorphic to U3, 5 or U4, 6 . The following result completes the proof of the theorem in the second case.

Lemma (4.2) . If n is 3 or 4, then the matroid Un, n+2 has exactly six inequivalent representations over GF(5).

Proof. Every GF(5)-representation for Un, n+2 is equivalent to one of the T form [In | A] where the columns of A are [1, 1, ..., 1] and [1, a1 , T a2 , ..., an&1] and a1 , a2 , ..., an&1 are distinct members of [2, 3, 4]. When n T is 3 or 4, there are six different choices for the column [1, a1 , a2 , ..., an&1] subject to these restrictions. It is straightforward to check that each such choice gives a representation for Un, n+2 and that different such choices give inequivalent representations. The lemma follows immediately. K

We may now assume that M has a U2, 5 -minor. If r(M)=2, then, since the automorphism group of PG(1, 5) is the symmetric group on six letters, all GF(5)-representations of M are equivalent. Hence we may assume that r(M)3. Thus, by Oxley [11, Theorem 2.1] (see also [12, Exercise 11.2.15(i)]), M is uniform, or M has a minor isomorphic to one of the matroids P6 and Q6 shown in Fig. 2. If M is uniform, then, by a result of Segre [15] (see also [12, Table 1, p. 206]), M$U3, 6 .IfMis not uniform, then, by the Splitter Theorem

[18] (see also [12, Chapter 11]), either M is isomorphic to P6 or Q6 ,or Mhas an element x such that M"x or MÂx is 3-connected and has a

P6 -orQ6-minor. We conclude that either

(i) M is isomorphic to U3, 6 , P6 ,orQ6;or (ii) M has an element x such that M"x or M*"x is 3-connected and has a P6 -orQ6-minor.

File: 582B 169511 . By:CV . Date:31:07:96 . Time:15:10 LOP8M. V8.0. Page 01:01 Codes: 3306 Signs: 2634 . Length: 45 pic 0 pts, 190 mm 336 OXLEY, VERTIGAN, AND WHITTLE

Figure 2

The next lemma completes the proof in case (i).

Lemma (4.3) . Each of the matroids U3, 6 , P6 , and Q6 has at most six inequivalent representations over GF(5).

Proof. Let N be in [U3, 6, P6 , Q6] where E(N)=[1, 2, ..., 6], with the labelling of P6 and Q6 being as in Fig. 2. Evidently N"6$U3, 5. We shall now show that, for each choice of N,

(4.4) every fixed representation for N"6 can be extended to a representation for N in at most one way.

While proving this last assertion, we shall view a representation for N"6 as a restriction of PG(2, 5). Clearly (4.4) holds when N=Q6 for, in that case, the point 6 must lie on the intersection of the lines of PG(2, 5) that are spanned by [2, 3] and [4, 5].

If N=P6 , then 6 must lie on the 6-point line of PG(2, 5) that is spanned by [4, 5]. Five of the points on this line are 4, 5, and the points of inter- section of L with the lines spanned by [1, 2], [1, 3], and [2, 3]. This leaves at most one choice for 6, so (4.4) holds when N=Q6 . Now suppose that N=U3, 6 . To establish (4.4) in this case, we shall assume, to the contrary, that there are two distinct points p1 and p2 that can be added to the fixed representation for N"6 to give a representation for N.By[15],U3, 7 is not GF(5)-representable, but PG(2, 5) | [1, 2, 3, 4, 5, pi]$U3, 6 for i=1, 2. Thus PG(2, 5) | [1, 2, 3, 4, 5, p1 , p2] has a unique line with more than two points. Without loss of generality, we may assume that this line is [p1,p2,5]. Let L be the line of PG(2, 5) containing these three points and let q1 , q2 , and q3 be the other three points on L. Clearly if [i, j] is a subset of [1, 2, 3, 4], then the line of PG(2, 5) spanned by [i, j] must meet L in q1 , q2 ,orq3. It now follows by Lemma 3.9 that the restriction of PG(2, 5) to [1, 2, 3, 4, q1 , q2 , q3] is isomorphic to F7 . This contradiction to the fact that F7 is only representable over fields of characteristic two completes the proof that (4.4) holds for N=U3, 6 , thereby finishing the proof of Lemma 4.3. K

File: 582B 169512 . By:SD . Date:28:06:96 . Time:07:57 LOP8M. V8.0. Page 01:01 Codes: 2753 Signs: 1966 . Length: 45 pic 0 pts, 190 mm ON INEQUIVALENT REPRESENTATIONS 337

We may now assume that (ii) holds. Thus M has an element x so that, for some N in [M, M*], the matroid N"x is 3-connected, has rank and corank at least three, and has a U2, 5-minor. The next lemma establishes that every representation for N"x extends in at most one way to a representation for N. By the choice of M, it follows from this lemma that N"x has at most six inequivalent GF(5)-representations. Hence, so does M; a contradiction. We conclude that the proof of Theorem 4.1 will be com- pleted once we have proved the next lemma.

(4.5) Lemma. Let T be a spanning subset of PG(r&1, 5) and suppose that PG(r&1, 5) | Tis3-connected and has a U2, 5-minor. If y1 and y2 are distinct elements of E(PG(r&1, 5))&T, then the map | that fixes each element of T and takes y1 to y2 is not an between PG(r&1, 5)|

(T_y1) and PG(r&1, 5) | (T_y2). Proof. Assume, to the contrary, that | is an isomorphism. Let L be the

6-point line of PG(r&1, 5) that is spanned by [y1,y2], let X=L& [y1,y2], and consider PG(r&1, 5) | (T_X). This matroid, M1 ,is certainly 3-connected and has a U2, 5-minor. Moreover, M1 | X$U2, 4 .It now follows by Corollary 3.10 that M1 has a U2, 5-minor using X. Let this minor be M1"VÂU for some independent set U and coindependent set V. Evidently r(U)=r(M1)&2. Let z be the element of M1"VÂU that is not in X.

Then U _ z spans a hyperplane of M1 . Moreover, this hyperplane avoids X. Thus U _ z spans a hyperplane H of PG(r&1, 5) that avoids X.In PG(r&1, 5), the line L and the hyperplane H must meet. Since H avoids

X, it contains exactly one of y1 and y2 . Without loss of generality, we may assume that y1 # H,soy2ÂH. Thus, in PG(r&1, 5), there is a circuit C containing y1 so that C&y1 U _ z T. Hence C is dependent, but |(C), which equals (C&y1) _ y2 , is not. This contradicts the assumption that | is an isomorphism. K

5. THE COUNTEREXAMPLES

To show that, for all prime powers q7, the number of inequivalent GF(q)-representations of a 3-connected matroid M is not bounded by some constant n(q), we shall consider two classes of examples. The first of these classes will establish the assertion for all prime powers q that exceed 5 and are not of the form 2 p where 2 p&1 is prime. Thus assume that q satisfies these conditions. The reason for imposing such conditions will be clear from the description of the example which we now give. Consider the multiplicative group GF(q)* of non-zero elements of GF(q). This group is cyclic and has q&1 elements. The choice of q guarantees that

File: 582B 169513 . By:CV . Date:31:07:96 . Time:15:10 LOP8M. V8.0. Page 01:01 Codes: 3117 Signs: 2504 . Length: 45 pic 0 pts, 190 mm 338 OXLEY, VERTIGAN, AND WHITTLE

GF(q)* has a proper subgroup A with at least three elements. For all r3, let Mr be the rank-r matroid that is represented by the matrix [Ir | D] where the columns of the identity matrix are labelled by e1 , e2 , ..., er ; and D is

f1 g1 f2 g2 f3 g3 }}} fr&1 gr&1 fr gr 110000 0 0 11

1:1 1100 0 0 00

001:2 11 0 0 00

00001:3 0000 bbbbbb}}} bbbb 000000 1 1 00 000000 1 : ; ; _ r&1 1 2& where each of :1 , :2 , ..., :r&1 is in A&[1], and ;1 , ;2 are distinct elements of GF(q)* that are not in the coset (&1)r A of the subgroup A. It will follow from the next result that Mr can be obtained from the rank-r whirl r W by freely adding a point on each dependent line, and hence Mr is certainly 3-connected.

Lemma (5.1) . The matroid Mr does not depend on the particular choice of the (r+1)-tuple (:1 , :2 , ..., :r&1, ;1, ;2).

Proof. It suffices to show that the non-spanning circuits of Mr do not depend on the choice of (:1 , :2 , ..., :r&1, ;1, ;2). For all i in [1, 2, ..., r], let

Ci*=[fi&1, gi&1, ei, fi, gi] where all subscripts are interpreted modulo r. Evidently C i* is a cocircuit of Mr . Moreover, for all j  [i&1, i, i+1], the matroid Mr"C*i Âej is disconnected although Mr"C i* is connected. It is straightforward to show using this that Mr"C i* can be constructed as follows where, as before, all subscripts are read modulo r. Begin with a

4-point line on [ei+1, fi+1, gi+1, ei+2]. Take the parallel connection of this matroid with a 4-point line on [ei+2, fi+2, gi+2, ei+3] using ei+2 as the basepoint. Then take the parallel connection of the resulting matroid with a 4-point line on [ei+3, fi+3, gi+3, ei+4], this time using ei+3 as the basepoint. Continue in this way and conclude by taking the parallel con- nection of [ei&2, fi&2, gi&2, ei&1] with the previously constructed matroid using ei&2 as the basepoint.

Since, for all i, the matroid Mr"C i* has the structure just described, this matroid does not depend on (:1 , :2 , ..., :r&1, ;1, ;2). Thus the only non- spanning circuits of Mr that could depend on this (r+1)-tuple are those that meet C*i for all i. Let C be such a circuit. Certainly |C|r. Moreover, as C meets C i* for all i, it follows that |C & C i*|2 for all i. Thus there are

File: 582B 169514 . By:CV . Date:31:07:96 . Time:15:10 LOP8M. V8.0. Page 01:01 Codes: 3227 Signs: 2124 . Length: 45 pic 0 pts, 190 mm ON INEQUIVALENT REPRESENTATIONS 339

at least 2r pairs (x, C*)i such that x # C & C*i and 1ir. Since no ele- ment of Mr is in more than two of the cocircuits C*1 , C*2 , ..., C*r and |C|r, it follows that there are at most 2r such pairs. Hence there are exactly 2r such pairs, |C|=r, and each element of C is in exactly two of the sets

C*1 , C*2 , ..., C*r . Thus C [f1,g1,f2,g2, ..., fr , gr]. Moreover, if C contains

[fi,gi]for some i, then C must avoid [fi+1, gi+1];soCmust also contain [fi+2, gi+2] and avoid [fi+3, gi+3], and so on. It then follows that C spans [e1, e2, ..., er]; a contradiction. We conclude that C contains exactly one element of [fj,gj] for all j in [1, 2, ..., r]. Therefore the matrix whose columns are the elements of C is

100 0 1

#1 10 0 0

0#2 100

00#3 00 bbb}}} bb 000 1 0 000 # ; _r&1 i& where #j # [1, :j] for all j in [1, 2, ..., r&1], and i # [1, 2]. Expanding down the last column, we see that this matrix has determinant equal to r&1 (&1) #1#2 }}}#r&1+;i. But this determinant is zero, and therefore ;i= r (&1) #1#2 }}}#r&1. This is a contradiction, for #1 #2 }}}#r&1 is certainly in r the subgroup A, but ;i was chosen not to lie in the coset (&1) A.We conclude that no circuit of Mr depends on (:1 , :2 , ..., :r&1, ;1, ;2). K

Proposition r (5.2) . The matroid Mr has at least 2 inequivalent represen- tations over GF(q).

Proof. In the matrix [Ir | D] representing Mr , the first non-zero entry of each row and column of D is a one. It follows without difficulty that the only way two different choices of (:1, :2 , ..., :r&1, ;1, ;2) can give equiv- alent representations for Mr is if the elements of one (r+1)-tuple can be obtained from the elements of the other by applying a fixed automorphism of GF(q). Now GF(q) has at most log2 q automorphisms, so the number of &1 inequivalent representations for Mr is at least (log2 q) times the number of choices for (:1 , :2 , ..., :r&1, ;1, ;2). The result now follows easily since each of :1 , :2 , ..., :r&1 can be chosen in at least two ways, and (;1 , ;2) can be chosen in at least ((q&1)Â2)((q&1)Â2&1) ways. K

The above construction for Nr makes no mention of Dowling group geometries [6]. Nonetheless the example was originally discovered using these matroids. For readers familiar with Dowling geometries, the following

File: 582B 169515 . By:CV . Date:31:07:96 . Time:15:10 LOP8M. V8.0. Page 01:01 Codes: 3309 Signs: 2203 . Length: 45 pic 0 pts, 190 mm 340 OXLEY, VERTIGAN, AND WHITTLE comments may add insight. The fact that A is a subgroup of the multi- plicative group of GF(q) means that Qr(A), the rank-r Dowling geometry over A, is representable over GF(q). Indeed, all the elements of Mr except fr and gr lie in the natural GF(q)-representation for Qr(A). The choice of fr and gr outside of this representation guarantees that no unwanted dependencies arise in Mr . To complete the proof that, for all q7, the number of inequivalent GF(q)-representations of a 3-connected matroid is not bounded by some constant n(q), we need to look at a second class of examples to treat the case when q=2t where t3 and q&1 is prime. The example will not need all these restrictions on q. Thus assume that q=pk for some prime p and some k2 where, if p=2, then k3. Let A be a proper additive subgroup of GF(q) having at least three elements. For all r4, let Nr be the rank-r matroid that is represented over GF(q) by the matrix [Ir | D] where the columns of the identity matrix are labelled by e1 , e2 , ..., er&1, g; and D is

f1 f2 f3 }}} fr&1 er fr 100 0 11 010 0 11 001 0 11 bbb}}} bbb 000 1 11 ::: : ; ; _1 2 3 r&1 1 2& where :1 , :2 , ..., :r&1 are non-zero elements of A, and ;1 and ;2 are distinct elements of GF(q)&A.

It is interesting to note that the matroid Nr depends on the additive structure of the field GF(q), whereas Mr depended on the multiplicative structure of the field. In spite of this difference, the proof of the next lemma is quite similar to the proof of Lemma 5.1.

Lemma (5.3) . The non-spanning circuits of Nr are the sets [g, ek , fk] with k in [1, 2, ..., r], and the sets [ei , ej , fi , fj] such that i and j are distinct elements of [1, 2, ..., r]. Hence Nr does not depend on the particular choice of (:1 , :2 , ..., :r&1, ;1, ;2).

Proof. The matroid Nr Âg can be obtained from an r-element circuit on [e1, e2 , ..., er] by adding fk in parallel with ek for all k. Thus, for all

2-element subsets [s, t] of [1, 2, ..., r], the set [es, et, fs, ft] is a cocircuit of Nr Âg and hence of Nr . Moreover, Nr Âg"[es, et, fs, ft] is the direct sum of r&2 two-element circuits. It follows easily that Nr "[es, et, fs, ft] is

File: 582B 169516 . By:CV . Date:31:07:96 . Time:15:10 LOP8M. V8.0. Page 01:01 Codes: 3074 Signs: 2090 . Length: 45 pic 0 pts, 190 mm ON INEQUIVALENT REPRESENTATIONS 341 obtained by taking the parallel connection of the r&2 three-point lines

[g, ek, fk] for which k # [1, 2, ..., r]&[s, t]. Hence, as r4, each of the sets specified in the statement of the lemma is a non-spanning circuit of Nr .

Suppose that Nr has some other non-spanning circuit C. Then C must meet all of the sets [es, et, fs, ft] where s and t are distinct elements of

[1, 2, ..., r]. Therefore, as each of these sets [es, et, fs, ft] is a cocircuit of

Nr and |C|r, it follows that C=[d1, d2, ..., dr] where dk # [ek, fk] for all k. Thus C&[e1, e2, ..., er&1] is a circuit of Nr Â(C & [e1, e2, ..., er&1]). Hence, for some subset [k1, k2, ..., km] of [1, 2, ..., r&1] and some i in [1, 2], the matrix

10 01 01 01 00 01 bb}}} bb 00 11 :: : ; _k1 k2 km i& has zero determinant. Thus the columns of this matrix are linearly dependent.

It follows easily that :k1+:k2 +}}}+:km=&;i. But the element on the left-hand side of this equation is in A, whereas &;i is certainly not. This contradiction completes the proof that the only non-spanning circuits of Nr are as specified in the lemma. K

It follows without difficulty from the last lemma that the matroid Nr is 3-connected.

Proposition r&1 (5.4) . The matroid Nr has at least 2 inequivalent representations over GF(q).

Proof. In the matrix [Ir | D] representing Nr , multiply the last row of &1 D by :1 to get D$. The matrix [Ir | D$] still represents Nr and this matrix has the maximum number of entries that, by row and column scaling, can be predetermined to equal one (see, for example, [12, Section 6.4]). Now argue as in the proof of Proposition 5.2. Each of :2 , :3 , ..., :r&1 can be chosen in at least two ways since |A|3. Moreover, (;1 , ;2) can be chosen in at least (qÂ2)(qÂ2&1) ways. Thus the number of inequivalent representa- r&2 k&1 k&1 &1 tions for Nr is at least 2 p (p &1) k since logp q=k. The proposition now follows easily. K We now consider the examples which establish Proposition 3.11. For the first example, we assume that q is an odd prime power exceeding 5. Since GF(q)* is cyclic of even order, it has a subgroup A of order (q&1)Â2.

File: 582B 169517 . By:CV . Date:31:07:96 . Time:15:10 LOP8M. V8.0. Page 01:01 Codes: 3020 Signs: 2053 . Length: 45 pic 0 pts, 190 mm 342 OXLEY, VERTIGAN, AND WHITTLE

T Define sets of vectors over GF(q) as follows: S1=[[1, :,0] ::#A]; S2= T T T T [[1,0,&:] : :#A]; S3=[[0, 1, 0] , [0, 0, 1] ] _ [[0, 1, :] : : # A]; T and, finally, S4 consists of all but two of the points of the form [0, 1, x] that are not in S3 . The fact that q7 guarantees that S4 is non-empty. Let MA be the matroid represented over GF(q)byS1_S2_S3_S4. Evidently,

M1 |(S3_S4)$U2, q&1. Moreover, it is straightforward to verify that MA has a U2, q -minor, and that MA has no U2, q -minor using S3 _ S4 . For readers familiar with Dowling geometries, the above verifications are particularly easy. The sets S1 , S2 , and S3 have been chosen so that T [[1,0,0] ]_S1_S2_S3 is a representation over GF(q) of the rank-3 Dowling geometry over A. This guarantees that, for a # S1 and b # S2 , the line clMA[a, b] meets the line S3 _ S4 , and does so in an element of S3 .It follows that the only non-trivial line passing through an arbitrary point p t of S is the line S _ S , and therefore M Âp $U . It also follows that if 4 t3 4 A 2, q x # S1 _ S2 , then MA Âx $U2, q&1. We conclude that MA has no U2, q-minor using S3 _ S4 . Now assume that q=2k, where k>2. Then the additive group of GF(q) has even order, and, since this group is abelian, it has a subgroup A of order qÂ2. In this case, define sets of vectors over GF(q)as T T T follows: S1=[[1, 0, :] : : # A]; S2=[[0, 1, :] : : # A]; S3=[[1, 1, :] : T : # A]_[[0,0,1] ]; and S4 consists of all but two points of the set T [[1, 1, ;] : ; Â A]. The fact that q8 guarantees that S4 is non-empty. Let NA be the matroid represented over GF(q)byS1_S2_S3_S4.

Clearly NA |(S3_S4)$U2, q&1. It is routine to verify that, for a # S1 and b # S , the line cl [a, b] meets the line S _ S , and does so in an element 2 NA 3 4 t of S3 . It follows from this that if p # S4 , then NA Âp $U2, q+1,soNA certainly has a U -minor. It also follows that if x # S _ S , then t 2, q 1 2 NA Âx $U2, q&1. We conclude that NA has no U2, q-minor using S3 _ S4 . Finally, it is worth noting that, just as with the matroids Mr and Nr defined before, the matroids MA and NA are constructed using properties of the multiplicative and additive groups of the field, respectively.

ACKNOWLEDGMENT

This research was partially supported by a grant from the Louisiana Education Quality Support Fund through the Board of Regents and by a grant from the National Security Agency.

REFERENCES

1. R. E. Bixby, On Reid's characterization of the ternary matroids, J. Combin. Theory Ser. B 26 (1979), 174204. 2. R. E. Bixby, A simple theorem on 3-connectivity, Appl. 45 (1982), 123126.

File: 582B 169518 . By:CV . Date:31:07:96 . Time:15:10 LOP8M. V8.0. Page 01:01 Codes: 3480 Signs: 2495 . Length: 45 pic 0 pts, 190 mm ON INEQUIVALENT REPRESENTATIONS 343

3. T. H. Brylawski, Modular constructions for combinatorial geometries, Trans. Amer. Math. Soc. 203 (1975), 144. 4. T. H. Brylawski and D. Lucas, Uniquely representable combinatorial geometries, in ``Teorie Combinatorie, Proc. 1973 Internat. Colloq.,'' pp. 83104, Accademia Nazionale dei Lincei, Rome, 1976. 5. C. R. Coullard and T. J. Reid, Element subsets of 3-connected matroids, Congress. Numer. 66 (1988), 8192. 6. T. A. Dowling, A class of geometric lattices based on finite groups, J. Combin. Theory Ser. B 14 (1973), 6186; erratum, 15 (1973), 211. 7. J. Geelen, A. M. H. Gerards, and A. Kapoor, in preparation. 8. J. Kahn, A geometric approach to forbidden minors for GF(3), J. Combin. Theory Ser. A 37 (1984), 112. 9. J. Kahn, On the uniqueness of matroid representations over GF(4), Bull. London Math. Soc. 20 (1988), 510. 10. J. Kahn and P. D. Seymour, On forbidden minors for GF(3), Proc. Amer. Math. Soc. 102 (1988), 437440. 11. J. G. Oxley, A characterization of certain excluded-minor classes of matroids, Europ. J. Combin. 10 (1989), 275279. 12. J. G. Oxley, ``Matroid Theory,'' Oxford Univ. Press, New York, 1992. 13. T. J. Reid, ``On Roundedness in Matroid Theory,'' Ph.D. dissertation, Louisiana State University, 1988. 14. G.-C. Rota, Combinatorial theory, old and new, in ``Proc. Internat. Cong. Math., Nice, 1970,'' pp. 229233, Gauthier-Villars, Paris, 1971. 15. B. Segre, Curve razionali normali e k-archi negli spazi finiti, Ann. Mat. Pura. Appl. 39 (1955), 357379. 16. P. D. Seymour, A note on the production of matroid minors, J. Combin. Theory Ser. B 22 (1977), 289295. 17. P. D. Seymour, Matroid representation over GF(3), J. Combin. Theory Ser. B 26 (1979), 159173. 18. P. D. Seymour, Decomposition of regular matroids, J. Combin. Theory Ser. B 28 (1980), 305359. 19. P. D. Seymour, On minors of non-binary matroids, Combinatorica 1 (1981), 387394. 20. P. D. Seymour, Minors of 3-connected matroids, Europ. J. Combin. 6 (1985), 375382. 21. K. Truemper, Alpha-balanced graphs and matrices and GF(3)-representability of matroids, J. Combin. Theory Ser. B 32 (1982), 112139. 22. W. T. Tutte, A homotopy theorem for matroids, I, II, Trans. Amer. Math. Soc. 88 (1958), 144174. 23. G. P. Whittle, Inequivalent representations of ternary matroids, Discrete Math. 149 (1996), 233238.

File: 582B 169519 . By:CV . Date:31:07:96 . Time:15:11 LOP8M. V8.0. Page 01:01 Codes: 2969 Signs: 2330 . Length: 45 pic 0 pts, 190 mm